Prpertatinn and Standard Deviation of Random Variable Moment Generating Function Engineering Assignment Help

Assignment Task

Task

Q- A bag contains 9 balls of identical shape and size. Each ball is identified with a unique number written on it from 1 to 9. Using the present setup, you perform an experiment in which you draw a ball randomly from the bag, note down the number written on it and put the ball back to the bag. Suppose, in the first try you draw a ball and got the number n. Now, you perform the experiment n more number of times and note down the number in each trial. Z is a random variable that records the sum of the numbers obtained in total n +1 trials including the find trial. (a) (6 points) Find the moment generating function of Z. (h) (3+3 points) Also compute the Prpertatinn and standard deviation of random variable 2. (10 points) Given a random variable X. Suppose we want to derive a bound similar to Chebyshev inequality, but using higher order moments: Hs[X] = Ef(X - EIX))1, where k > 2. Note that Chebyshev inequality uses 2nd order moments. ill X (a) (3 points) Show that P(IX — EIXII > + < , where 6 > 0 and k is even. Why is it difficult to derive a bound for an odd value of k? Justify. (b) (3 points) Find probability bound for the event P(p- Va-o < X xss=removed xss=removed>  

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